Abstract
The current study is devoted to deriving some results about existence and stability analysis for a nonlinear problem of implicit fractional differential equations (FODEs) with impulsive and integral boundary conditions. The concerned problem involves proportional type delay term. By using Schaefer’s fixed point theorem and Banach’s contraction principle, the required conditions are developed. Also, different kinds of Ulam stability results are derived by using nonlinear analysis. Providing a pertinent example, we demonstrate our main results.
Highlights
1 Introduction In many real world problems, the fractional order models are found to be more suitable than integer order ones
We can find the applications of fractional order derivatives and integrals in electrodynamics of complex medium, aerodynamics, polymer rheology, physics, chemistry, and so forth
Problems with integral boundary conditions naturally arise in applied fields of science like thermal conduction problems, semiconductor problems, chemical engineering, blood flow problems, underground water flow problems, hydrodynamic problems, population dynamics, and so forth
Summary
In many real world problems, the fractional order models are found to be more suitable than integer order ones. Definition 2.4 ([40]) If for > 0 there exists a constant Cg > 0 such that, for any solution ψ ∈ W of inequality (5), there is a unique solution υ ∈ W of system (1) that satisfies ψ(t) – υ(t) ≤ Cg , t ∈ I, problem (1) is Hyers–Ulam stable. Definition 2.6 ([40]) If for > 0 there exists a real number Cg > 0 such that, for any solution ψ ∈ W of inequality (7), there is a unique solution υ ∈ W of problem (1) that satisfies ψ(t) – υ(t) ≤ Cg ξ + x(t) , t ∈ I, problem (1) is Hyers–Ulam–Rassias stable with respect to (ξ , x). To prove the result, we further need another assumption as follows: (H8) Let, for a nondecreasing function x ∈ C(I, R), there exist constants μx > 0, > 0 such that.
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